We report the first measurement of the differential cross section for the process gamma gamma -> eta eta in the kinematic range above the eta eta threshold, 1.096 GeV < W < 3.8 GeV over nearly the entire solid angle range, vertical bar cos theta*vertical bar <= 0.9 or <= 1.0 depending on W, where W and theta* are the energy and eta scattering angle, respectively, in the gamma gamma center-of-mass system. The results are based on a 393 fb(-1) data sample collected with the Belle detector at the KEKB e(+)e(-) collider. In the W range 1.1-2.0 GeV/c(2) we perform an analysis of resonance amplitudes for various partial waves, and at higher energy we compare the energy and the angular dependences of the cross section with predictions of theoretical models and extract contributions of the chi(cJ) charmonia.